Each digit, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter A, B, C, D, E, F, G, H and I but not necessarily in this order. Further, each of A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
Determine the value of each of the alphabets such that the above conditions are satisfied.

A survey is conducted in a school among 150 students and it was found that 100 students play cricket, 90 students play football and 80 students play hockey. As physical education was given emphasis in the school, every student had to play at least one sport.
What can be the maximum number of students who play exactly two sports?
What can be the maximum number of students who play exactly one sport?

Prove or Disprove:

There exists some power of 3 which ends in 001.

How many 5 digit numbers are there such that the sum of its digits is odd?
Example: 12345 is a five digit number such that the sum of its digits = 1 + 2 + 3 + 4 + 5 = 15 = an odd number.

Prove that:
"In any group of 6 people, there are either 3 mutual friends or 3 mutual strangers".

Mr. A, Mr. B, Mr. C and Mr. D are four men who are Physicist, Mathematician, Doctor and Policeman, though not necessarily in that order. Mr. A and Mr. B are neighbours and they go together to their work by a bike. Mr. B makes more money than Mr. C. Mr. A meets Mr. D regularly in chess playing. The Physicist always goes by car. The only time the Doctor met the Policeman ever was when the Policeman arrested the Doctor for speeding. The policeman makes more money than the Mathematician. The Doctor makes more money than Mr. D

Who is the Physicist?
(a) Mr. A
(b) Mr. C
(c) Mr. D
(d) Mr. B
(e) Cannot be determined

Who is the second highest earner?
(a) Mr. C
(b) Mr. D
(c) Mr. A
(d) Either (a) or (b)
(e) Cannot be determined

Who is the Mathematician?
(a) Mr. A
(b) Mr. D
(c) Mr. B
(d) Mr. C
(e) Cannot be determined

What is the profession of Mr. B?
(a) Physicist
(b) Policeman
(c) Mathematician
(d) Doctor
(e) Cannot be determined

Let us suppose that there are 100 students in my class. Prove that among them there are at least two of them who have an equal number of friends.
Note: If A is friends with B then it also implies that B is friends with A or else A and B both are strangers to each other.

In a street there are 5 houses painted 5 different colours. In each house lives a person of a different nationality. The 5 homeowners each drink a different beverage, smoke a different brand of cigar, and keep a different pet.
1. The Brit lives in the Red house.
2. The Swede has a Dog.
3. The Dane drinks Tea.
4. The Green house is on the left of the White house.
5. The owner of the Green house drinks Coffee.
6. The person who smokes Pall Mall has Birds.
7. The owner of the Yellow house smokes Dunhill.
8. The man living in the center house drinks Milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blends lives next to the one who has Cats.
11. The man who has Horses lives next to the man who smokes Dunhill.
12. The man who smokes Blue Master drinks Beer.
13. The German smokes Prince.
14. The Norwegian lives next to the Blue house.
15. The man who smokes Blends has a neighbour who drinks water.

Find out the correct combination of drinks, pets, smokes and colour of the house for each of the five persons.

A very long hallway has 1000 doors numbered 1 to 1000; all doors are initially closed. One by one, 1000 people get into the hall; the first person opens each door, the second closes all door with even numbers, the third person closes door 3, opens door 6, closes door 9, opens door 12 etc. That is, the nth person changes all doors whose numbers are divisible by n. After all 1000 people have got into the hall, find out how many doors will remain open.

In the shooting of the movie, "A Beautiful Mind", Ravi, the actor, is running towards a vertical mirror with a speed of 10 m/sec on a line which is perpendicular to the plane of the mirror. The mirror is moving towards Ravi with a speed of 6 m/sec along the same line. Find the speed of Ravi’s image with respect to Ravi (the image of Ravi forms on the other side of the mirror such that the distance between the image of Ravi and the mirror is equal to the distance between Ravi and the mirror).

There are 60 workers who work for a company, out of which 25 are women. Also:
28 workers are married
26 workers are graduate
20 married workers are graduate of which 9 are men
15 men are graduate
15 men are married
How many unmarried women are graduates and how many unmarried women work in the company?